Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $y \neq 0$. $q = \dfrac{7y - 63}{2y} \div \dfrac{2y - 18}{5y} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{7y - 63}{2y} \times \dfrac{5y}{2y - 18} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (7y - 63) \times 5y } { 2y \times (2y - 18) } $ $ q = \dfrac {5y \times 7(y - 9)} {2y \times 2(y - 9)} $ $ q = \dfrac{35y(y - 9)}{4y(y - 9)} $ We can cancel the $y - 9$ so long as $y - 9 \neq 0$ Therefore $y \neq 9$ $q = \dfrac{35y \cancel{(y - 9})}{4y \cancel{(y - 9)}} = \dfrac{35y}{4y} = \dfrac{35}{4} $